Baire irresolvable spaces and ideal theory
Abstract
In the New Scottish Book M. Katĕtov asked whether there exists a Hausdorff space X without isolated points such that every real-valued function on X is continuous at some point? In the paper it is shown that the existence of such a space is equiconsistent to the existence of measurable cardinal.
References
1. J. Baumgartner, A. Taylor, S. Wagon, Structural properties of ideals, (to appear in Dissertationes Math.).
2. A.G. Elkin, Ultrafilters and irresolvable spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 24 (1969), 51-56.
3. A.M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489.
4. E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333.
5. T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165-198.
6. T. Jech, Set Theory, Academic Press 1978.
7. T. Jech, M. Magidor, W. Mitchell, K. Prikry, Precipitous ideals, (to appear).
8. T. Jech, K. Prikry, Ideals over uncountable sets: Applications of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc. 214.
9. M. Katĕtov, On topological spaces containing no disjoint dense sets, Mat. Sb. 21 (1947), 3-12.
10. K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227.
11. K. Kunen, Saturated ideals, J. Symbolic Logic, 43 (1978), 65-76.
12. K. Kunen, Maximal ?-Independent Families, Technical Report # 15, Univer. of Texas at Austin, Dec. 1979.
13. D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987-994.
14. V.I. Malyhin, On the resolvability of the product of two spaces and a problem of Katětov, Dokl. Akad. Nauk SSSR 222 (1975), 725-729.
15. R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56.
16. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math. 13 (1971), 397-428.
17. A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 33-55.
18. A. Taylor, Diamond principles, ideals and the normal Moore space problem, Canad. J. Math., (to appear).
19. A. Ionescu Tulcea, C. Ionescu Tulcea, Topics in the theory of liftings, Ergebnisse der Math. 48, Springer-Verlag, Berlin 1969.
20. S. Wagon, The structure of precipitous ideals, Fund. Math. 106 (1980), 47-52.
21. B. Węglorz, Some properties of filters, (preprint).
22. H. Woodin, An ℵ_1-dense ℵ_1-complete ideal on ℵ_1, (preprint).
2. A.G. Elkin, Ultrafilters and irresolvable spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 24 (1969), 51-56.
3. A.M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489.
4. E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333.
5. T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165-198.
6. T. Jech, Set Theory, Academic Press 1978.
7. T. Jech, M. Magidor, W. Mitchell, K. Prikry, Precipitous ideals, (to appear).
8. T. Jech, K. Prikry, Ideals over uncountable sets: Applications of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc. 214.
9. M. Katĕtov, On topological spaces containing no disjoint dense sets, Mat. Sb. 21 (1947), 3-12.
10. K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227.
11. K. Kunen, Saturated ideals, J. Symbolic Logic, 43 (1978), 65-76.
12. K. Kunen, Maximal ?-Independent Families, Technical Report # 15, Univer. of Texas at Austin, Dec. 1979.
13. D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987-994.
14. V.I. Malyhin, On the resolvability of the product of two spaces and a problem of Katětov, Dokl. Akad. Nauk SSSR 222 (1975), 725-729.
15. R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56.
16. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math. 13 (1971), 397-428.
17. A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 33-55.
18. A. Taylor, Diamond principles, ideals and the normal Moore space problem, Canad. J. Math., (to appear).
19. A. Ionescu Tulcea, C. Ionescu Tulcea, Topics in the theory of liftings, Ergebnisse der Math. 48, Springer-Verlag, Berlin 1969.
20. S. Wagon, The structure of precipitous ideals, Fund. Math. 106 (1980), 47-52.
21. B. Węglorz, Some properties of filters, (preprint).
22. H. Woodin, An ℵ_1-dense ℵ_1-complete ideal on ℵ_1, (preprint).
KunenK., SzymańskiA., & TallF. (1986). Baire irresolvable spaces and ideal theory. Annales Mathematicae Silesianae, 2, 98-107. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14330
Kenneth Kunen
Department of Mathematics, University Wisconsin, U.S.A. United States
Department of Mathematics, University Wisconsin, U.S.A. United States
Andrzej Szymański
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Franklin Tall
Department of Mathematics, University of Toronto, Canada Canada
Department of Mathematics, University of Toronto, Canada Canada
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
